Pour enough oil into a 12-quart pot so that it reaches a depth of 1". Heat over medium-high heat until a deep-fry thermometer inserted in the oil registers 375°. Make a gremolata by mixing parsley, garlic, and lemon zest in a large bowl; set aside. Over a kitchen towel, julienne potatoes lengthwise using the narrowest setting of a French mandoline. Pat julienned potatoes dry with the towel.
Working in 2 batches, fry potatoes in oil until crisp, 6–10 minutes. Using a slotted spoon, transfer fries to a paper towel–lined bowl. Season with salt. (Bring oil to 375° before frying second batch.) Transfer fries to bowl of gremolata; toss to coat fries evenly. (Serve hot or at room temperature.)
Build a medium-hot charcoal fire in a grill or heat a gas grill to medium-high. Meanwhile, bring beer, onions, cloves, and 1 cup water to a boil in a 2-quart saucepan. Boil bratwurst in beer until almost cooked through, about 5 minutes. Drain.
Grill bratwurst, turning once, until browned, about 6 minutes. Using tongs, transfer bratwurst to a plate. Brush buns with the butter and grill until toasted and golden brown. Serve the bratwurst in the buns, slathered with mustard and garnished with the pickled cauliflower, alongside the fries.